Almost all

In mathematics, the phrase almost all has a number of specialised uses.

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"Almost all" is sometimes used synonymously with "all but finitely many"; see almost.

Related Topics:
Finite - Almost

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In number theory, if P(n) is a property of positive integers, and if p(N) denotes the number of positive integers n less than N for which P(n) holds, and if

Related Topics:
Number theory - Integer

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:p(N)/N → 1 as N → ∞

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(see limit), then we say that "P(n) holds for almost all positive integers n" and write

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:(orall^infty n) P(n).

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For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N/ln N. Therefore the proportion of prime integers is roughly 1/ln N, which tends to 0. Thus, almost all positive integers are composite.

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Occasionally, "almost all" is used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability theory.

Related Topics:
Almost everywhere - Measure theory - Almost surely - Probability theory

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